Complex manifolds

A complex manifold of complex dimension n is a special case of a real mani-

fold of dimension d = 2n. It is de¬ned in an analogous manner using complex

local coordinate systems. In this case the transition functions are required

to be biholomorphic, which means that they and their inverses are both

holomorphic. Let us denote complex local coordinates by z a (a = 1, . . . , n)

and their complex conjugates z a .

¯¯

A complex manifold admits a tensor J, with one covariant and one con-

travariant index, which in complex coordinates has components

¯ ¯ ¯

Ja b = iδa b , Ja b = ’iδa b , Ja b = Ja b = 0. (9.258)

¯ ¯ ¯

These equations are preserved by a holomorphic change of variables, so they

describe a globally well-de¬ned tensor.

Sometimes one is given a real manifold M in 2n dimensions, and one

wishes to determine whether it is a complex manifold. The ¬rst requirement

is the existence of a tensor, J m n , called an almost complex structure, that

satis¬es

Jm n Jn p = ’δm p . (9.259)

This equation is preserved by a smooth change of coordinates. The second

condition is that the almost complex structure is a complex structure. The

obstruction to this is given by a tensor, called the Nijenhuis tensor

N p mn = Jm q ‚[q Jn] p ’ Jn q ‚[q Jm] p . (9.260)

When this tensor is identically zero, J is a complex structure. Then it is

possible to choose complex coordinates in every open set that de¬nes the

real manifold M such that J takes the values given in Eq. (9.258) and the

transition functions are holomorphic.

On a complex manifold one can de¬ne (p, q)-forms as having p holomorphic

and q antiholomorphic indices

1 ¯ ¯

Aa1 ···ap¯1 ···¯q dz a1 § · · · § dz ap § d¯b1 § · · · § d¯bq .

Ap,q = z z (9.261)

bb

p!q!

The real exterior derivative can be decomposed into holomorphic and anti-

holomorphic pieces

¯

d=‚+‚ (9.262)

Appendix: Some basic geometry and topology 449

with

‚ ‚

¯

‚ = dz a ‚ = d¯a a .

z¯ ¯

and (9.263)

‚z a ‚z

¯

¯

Then ‚ and ‚, which are called Dolbeault operators, map (p, q)-forms to

(p + 1, q)-forms and (p, q + 1)-forms, respectively. Each of these exterior

derivatives is nilpotent

¯

‚ 2 = ‚ 2 = 0, (9.264)

and they anticommute

¯¯

‚ ‚ + ‚‚ = 0. (9.265)

Complex geometry

Let us now consider a complex Riemannian manifold. In terms of the com-

plex local coordinates, the metric tensor is given by

¯ ¯

ds2 = gab dz a dz b + ga¯dz a d¯b + gab d¯a dz b + ga¯d¯a d¯b .

¯ ¯

z ¯z ¯b z z (9.266)

b

The reality of the metric implies that ga¯ is the complex conjugate of gab

¯b

and that ga¯ is the complex conjugate of gab . A hermitian manifold is a

¯

b

special case of a complex Riemannian manifold, which is characterized by

the conditions

gab = ga¯ = 0. (9.267)

¯b

These conditions are preserved under holomorphic changes of variables, so

they are globally well de¬ned.

p,q

The Dolbeault cohomology group H‚ (M ) of a hermitian manifold M

¯

¯

consists of equivalence classes of ‚-closed (p, q)-forms. Two such forms are

¯

equivalent if and only if they di¬er by a ‚-exact (p, q)-form. The dimension

p,q

of H‚ (M ) is called the Hodge number hp,q . We can de¬ne the Laplacians

¯

¯¯ ¯¯

∆‚ = ‚‚ † + ‚ † ‚ ∆‚ = ‚ ‚ † + ‚ † ‚.

and (9.268)

¯

A K¨hler manifold is de¬ned to be a hermitian manifold on which the

a

K¨hler form

a

¯

J = iga¯dz a § d¯b

z (9.269)

b

is closed

dJ = 0. (9.270)

It follows that the metric on these manifolds satis¬es ‚a gb¯ = ‚b ga¯, as well

c

c

450 String geometry

as the complex conjugate relation, and therefore it can be written locally in

the form

‚‚

ga¯ = a ¯ K(z, z), ¯ (9.271)

b

‚z ‚ z b

¯

where K(z, z ) is called the K¨hler potential. Thus,

¯ a

¯

J = i‚ ‚K.

The K¨hler potential is only de¬ned up to the addition of arbitrary holo-

a

¯z

morphic and antiholomorphic functions f (z) and f (¯), since

¯z

˜¯

K(z, z ) = K(z, z ) + f (z) + f (¯)

¯ (9.272)

leads to the same metric. In fact, there are such relations on the overlaps of

open sets.

On K¨hler manifolds the various Laplacians all become identical

a

∆d = 2∆‚ = 2∆‚ . (9.273)

¯

¯

The various possible choices of cohomology groups (based on d, ‚ and ‚)

each have a unique harmonic representative of the corresponding type, as in

the real case described earlier. Therefore, in the case of K¨hler manifolds,

a

it follows that they are all identical

p,q p,q